[/quote]
,
that satisfy the equation
.
and 
be points on the side BC such that 
and 
,
and 
be the incenter of triangles 
and 
,
is the Euler line of triangle 
.
,
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
be reals such that 
and 
Prove that
be a cyclic quadrilateral. The point 
the intersection of 
.
.
concyclic.
. Prove that: 
and ray 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle 
distinct from point 
and 
intersect 
and 
respectively.
such that 
for all real number 
and 
is an equilateral . Let 
and 
at 
and 
at 
,
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds.
be a positive integer such that 
.
.
Y by
Let 
be a triangle with circumcircle 
. 
is the midpoint of arc 
(this arc does not contain 
). 
is the common point of and the perpendicular bisector of 
. 
is the common point of 
and the parallel to 
containing . 
is the common point of 
and . 
is the common point of 
and . Show that 
is isosceles.
be a triangle with circumcircle 
. 
is the midpoint of arc 
(this arc does not contain 
). 
is the common point of and the perpendicular bisector of 
. 
is the common point of 
and the parallel to 
containing . 
is the common point of 
and . 
is the common point of 
and . Show that 
is isosceles.
we get that 
is cyclic (the circumcenter of 
also lies on this circle, but that's not important in this problem). Thus, we also have
whence 
is cyclic as well. However, 
not containing 
so 
bisects 
but 
and 
too, so we're done. 
